Optimal. Leaf size=25 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
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Rubi [A] time = 0.0288314, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3190, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos (x)}{a+b \sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sin (x)\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0074584, size = 25, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 17, normalized size = 0.7 \begin{align*}{\arctan \left ({\sin \left ( x \right ) b{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03186, size = 190, normalized size = 7.6 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (-\frac{b \cos \left (x\right )^{2} + 2 \, \sqrt{-a b} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right )}{2 \, a b}, \frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b} \sin \left (x\right )}{a}\right )}{a b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.63744, size = 87, normalized size = 3.48 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sin{\left (x \right )}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{\sin{\left (x \right )}}{a} & \text{for}\: b = 0 \\- \frac{1}{b \sin{\left (x \right )}} & \text{for}\: a = 0 \\- \frac{i \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sin{\left (x \right )} \right )}}{2 \sqrt{a} b \sqrt{\frac{1}{b}}} + \frac{i \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sin{\left (x \right )} \right )}}{2 \sqrt{a} b \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.121, size = 22, normalized size = 0.88 \begin{align*} \frac{\arctan \left (\frac{b \sin \left (x\right )}{\sqrt{a b}}\right )}{\sqrt{a b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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